Turbulence is usually addressed with the aid of the kinetic energy spectrum whose dependence upon the wavenumber \(k\)  allows
to distinguish different regimes that are relevant for characterizing it.[1]} The kinetic energy of the superfluid is written as
\(E_{\textrm {kin}}(t) = \frac{\hbar ^2}{2m} \int {\textrm {d}}{\mathbf {r}} \, \left|\nabla \sqrt{\rho ({\mathbf {r}},t)}\right|^2+ \frac{m}{2} \int {\textrm {d}}{\mathbf {r}} \, \frac{{\mathbf {j}}^2({\mathbf {r}},t)}{\rho ({\mathbf {r}},t)}\; ,\) 
